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Eric Goubault

Publications and source records attributed to Eric Goubault.

At least 19 recordsLinked to original sources
arXiv2026

Propagation of~Interval Belief Structures and~Imprecise Copulas for~Neural Network Verification

Quantitative verification of neural networks requires reasoning about probabilities under substantial uncertainty in both input distributions and their dependence structure. In realistic settings, this information is often only partially specified, and assuming precise probabilistic models can lead to unreliable results. We propose a sound framework for quantitative verification under imprecise probabilistic information, combining interval belief structures to represent marginal uncertainty with imprecise copulas to model uncertain dependence. We develop a propagation method for imprecisely coupled interval belief structures through feed-forward neural networks. Using mixed imprecise copula volumes, we derive sound push-forward constructions through affine transformations and activation functions. The resulting output can provide guaranteed lower and upper bounds on probabilistic safety properties, valid for all probability models compatible with the specified imprecise inputs.

cs.AI↗
arXiv2026

Directed homological and cohomological operations

In this short note, we present a persistence module approach to directed cohomology, dual to the directed homology introduced by the author in a previous article. We lay out the first properties of directed cohomology and in particular of cohomological operations, partially linked to some homological operations. We treat here both the case of a specific class of precubical sets and of general directed spaces.

math.AT↗
arXiv2026

Homological Algebra in Abelian Framed Bicategories: Exact Sequences and Embedding Theorems

We introduce abelian framed bicategories, which are particular framed bicategories that are locally abelian, and show that they are suitable for developing homology and cohomology theories for directed structures. This means in particular that similar exact sequences as the relative homology and Mayer-Vietoris long exact sequences can be shown to hold. Also, for closed monoidal abelian framed bicategories, K\"unneth theorem holds as well. Finally, we prove embedding theorems similar to the Gabriel and Freyd-Mitchell theorems, for particular abelian framed bicategories, allowing to see those as bicategories of bimodules over algebras. This naturally links to the original motivation of this work, which was to generalize directed homology developed in the abelian framed bicategory of bimodules over (path) algebras.

math.CT↗
arXiv2025

A categorical and logical framework for iterated protocols

In this article, we show that the now classical protocol complex approach to distributed task solvability of Herlihy et al. can be understood in standard categorical terms. First, protocol complexes are functors, from chromatic (semi-) simplicial sets to chromatic simplicial sets, that naturally give rise to algebras. These algebras describe the next state operator for the corresponding distributed systems. This is constructed for semi-synchronous distributed systems with general patterns of communication for which we show that these functors are always Yoneda extensions of simpler functors, implying a number of interesting properties. Furthermore, for these protocol complex functors, we prove the existence of a free algebra on any initial chromatic simplicial complex, modeling iterated protocol complexes. Under this categorical formalization, protocol complexes are seen as transition systems, where states are structured as chromatic simplicial sets. We exploit the epistemic interpretation of chromatic simplicial sets and the underlying transition system (or algebra) structure to introduce a temporal-epistemic logic and its semantics on all free algebras on chromatic simplicial sets. We end up by giving hints on how to extend this framework to more general dynamic network graphs and state-dependent protocols, and give example in fault-tolerant distributed systems and mobile robotics.

cs.LO↗
arXiv2025

Directed homotopy modules

In this short note, we argue that directed homotopy can be given the structure of generalized modules, over particular monoids. This is part of a general attempt for refoundation of directed topology.

math.AT↗
arXiv2025

Knowledge in multi-robot systems: an interplay of dynamics, computation and communication

In this paper, we provide a framework integrating distributed multi-robot systems and temporal epistemic logic. We show that continuous-discrete hybrid systems are compatible with logical models of knowledge already used in distributed computing, and demonstrate its usefulness by deriving sufficient epistemic conditions for exploration and gathering robot tasks to be solvable. We provide a separation of the physical and computational aspects of a robotic system, allowing us to decouple the problems related to each and directly use methods from control theory and distributed computing, fields that are traditionally distant in the literature. Finally, we demonstrate a novel approach for reasoning about the knowledge in multi-robot systems through a principled method of converting a switched hybrid dynamical system into a temporal-epistemic logic model, passing through an abstract state machine representation. This creates space for methods and results to be exchanged across the fields of control theory, distributed computing and temporal-epistemic logic, while reasoning about multi-robot systems.

cs.LO↗
arXiv2023

Simplicial Models for the Epistemic Logic of Faulty Agents

In recent years, several authors have been investigating simplicial models, a model of epistemic logic based on higher-dimensional structures called simplicial complexes. In the original formulation, simplicial models were always assumed to be pure, meaning that all worlds have the same dimension. This is equivalent to the standard S5n semantics of epistemic logic, based on Kripke models. By removing the assumption that models must be pure, we can go beyond the usual Kripke semantics and study epistemic logics where the number of agents participating in a world can vary. This approach has been developed in a number of papers, with applications in fault-tolerant distributed computing where processes may crash during the execution of a system. A difficulty that arises is that subtle design choices in the definition of impure simplicial models can result in different axioms of the resulting logic. In this paper, we classify those design choices systematically, and axiomatize the corresponding logics. We illustrate them via distributed computing examples of synchronous systems where processes may crash.

cs.LO↗
arXiv2023

Guaranteed approximations of arbitrarily quantified reachability problems

We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.

eess.SY↗
arXiv2023

Directed Homology and Persistence Modules

In this note, we give a self-contained account on a construction for a directed homology theory based on modules over algebras, linking it to both persistence homology and natural homology. We study its first properties, among which some exact sequences.

math.AT↗
arXiv2023

Estimating the Coverage Measure and the Area Explored by a Line-Sweep Sensor on the Plane

This paper presents a method for determining the area explored by a line-sweep sensor during an area-covering mission in a two-dimensional plane. Accurate knowledge of the explored area is crucial for various applications in robotics, such as mapping, surveillance, and coverage optimization. The proposed method leverages the concept of coverage measure of the environment and its relation to the topological degree in the plane, to estimate the extent of the explored region. In addition, we extend the approach to uncertain coverage measure values using interval analysis. This last contribution allows for a guaranteed characterization of the explored area, essential considering the often critical character of area-covering missions. Finally, this paper also proposes a novel algorithm for computing the topological degree in the 2-dimensional plane, for all the points inside an area of interest, which differs from existing solutions that compute the topological degree for single points. The applicability of the method is evaluated through a real-world experiment.

cs.RO↗
arXiv2023

A many-sorted epistemic logic for chromatic hypergraphs

We propose a many-sorted modal logic for reasoning about knowledge in multi-agent systems. Our logic introduces a clear distinction between participating agents and the environment. This allows to express local properties of agents and global properties of worlds in a uniform way, as well as to talk about the presence or absence of agents in a world. The logic subsumes the standard epistemic logic and is a conservative extension of it. The semantics is given in chromatic hypergraphs, a generalization of chromatic simplicial complexes, which were recently used to model knowledge in distributed systems. We show that the logic is sound and complete with respect to the intended semantics. We also show a further connection of chromatic hypergraphs with neighborhood frames.

cs.LO↗
arXiv2023

Persistent homology of partially ordered spaces

In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was developed in the context of topological data analysis, in which topological properties of point-cloud data sets are extracted while eliminating noise. In both approaches, the evolution homological properties are tracked through a sequence of inclusions of usual topological spaces. Exploiting this similarity, we show that natural homology may be considered a persistence object, and may be calculated as a colimit of uni-dimensional persistent homologies along traces. Finally, we suggest further links and avenues of future work in this direction.

math.AT↗
arXiv2023

Semi-simplicial Set Models for Distributed Knowledge

In recent years, a new class of models for multi-agent epistemic logic has emerged, based on simplicial complexes. Since then, many variants of these simplicial models have been investigated, giving rise to different logics and axiomatizations. In this paper, we present a further generalization, where a group of agents may distinguish two worlds, even though each individual agent in the group is unable to distinguish them. For that purpose, we generalize beyond simplicial complexes and consider instead simplicial sets. By doing so, we define a new semantics for epistemic logic with distributed knowledge. As it turns out, these models are the geometric counterpart of a generalization of Kripke models, called "pseudo-models". We identify various interesting sub-classes of these models, encompassing all previously studied variants of simplicial models; and give a sound and complete axiomatization for each of them.

cs.LO↗
arXiv2023

A semi-abelian approach to directed homology

We develop a homology theory for directed spaces, based on the semi-abelian category of (non-unital) associative algebras. The major ingredient is a simplicial algebra constructed from convolution algebras of certain trace categories of a directed space. We show that this directed homology HA is invariant under directed homeomorphisms, and is computable as a simple algebra quotient for $HA_1$. We also show that the algebra structure for $HA_n$, $n\geq 2$ is degenerate, through a Eckmann-Hilton argument. We hint at some relationships between this homology theory and natural homology, another homology theory designed for directed spaces. Finally we pave the way towards some interesting long exact sequences.

math.AT↗
arXiv2022

Taylor-Lagrange Neural Ordinary Differential Equations: Toward Fast Training and Evaluation of Neural ODEs

Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every forward evaluation of a NODE requires numerical integration of the neural network used to capture the system dynamics, making their training prohibitively expensive. Existing works rely on off-the-shelf adaptive step-size numerical integration schemes, which often require an excessive number of evaluations of the underlying dynamics network to obtain sufficient accuracy for training. By contrast, we accelerate the evaluation and the training of NODEs by proposing a data-driven approach to their numerical integration. The proposed Taylor-Lagrange NODEs (TL-NODEs) use a fixed-order Taylor expansion for numerical integration, while also learning to estimate the expansion's approximation error. As a result, the proposed approach achieves the same accuracy as adaptive step-size schemes while employing only low-order Taylor expansions, thus greatly reducing the computational cost necessary to integrate the NODE. A suite of numerical experiments, including modeling dynamical systems, image classification, and density estimation, demonstrate that TL-NODEs can be trained more than an order of magnitude faster than state-of-the-art approaches, without any loss in performance.

cs.LG↗
arXiv2021

Solving N-player dynamic routing games with congestion: a mean field approach

The recent emergence of navigational tools has changed traffic patterns and has now enabled new types of congestion-aware routing control like dynamic road pricing. Using the fundamental diagram of traffic flows - applied in macroscopic and mesoscopic traffic modeling - the article introduces a new N-player dynamic routing game with explicit congestion dynamics. The model is well-posed and can reproduce heterogeneous departure times and congestion spill back phenomena. However, as Nash equilibrium computations are PPAD-complete, solving the game becomes intractable for large but realistic numbers of vehicles N. Therefore, the corresponding mean field game is also introduced. Experiments were performed on several classical benchmark networks of the traffic community: the Pigou, Braess, and Sioux Falls networks with heterogeneous origin, destination and departure time tuples. The Pigou and the Braess examples reveal that the mean field approximation is generally very accurate and computationally efficient as soon as the number of vehicles exceeds a few dozen. On the Sioux Falls network (76 links, 100 time steps), this approach enables learning traffic dynamics with more than 14,000 vehicles.

math.DS↗
arXiv2021

Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling

Effective inclusion of physics-based knowledge into deep neural network models of dynamical systems can greatly improve data efficiency and generalization. Such a-priori knowledge might arise from physical principles (e.g., conservation laws) or from the system's design (e.g., the Jacobian matrix of a robot), even if large portions of the system dynamics remain unknown. We develop a framework to learn dynamics models from trajectory data while incorporating a-priori system knowledge as inductive bias. More specifically, the proposed framework uses physics-based side information to inform the structure of the neural network itself, and to place constraints on the values of the outputs and the internal states of the model. It represents the system's vector field as a composition of known and unknown functions, the latter of which are parametrized by neural networks. The physics-informed constraints are enforced via the augmented Lagrangian method during the model's training. We experimentally demonstrate the benefits of the proposed approach on a variety of dynamical systems -- including a benchmark suite of robotics environments featuring large state spaces, non-linear dynamics, external forces, contact forces, and control inputs. By exploiting a-priori system knowledge during training, the proposed approach learns to predict the system dynamics two orders of magnitude more accurately than a baseline approach that does not include prior knowledge, given the same training dataset.

cs.LG↗